### robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Scaling SLAM Algorithms

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Covariance Intersection

SLAM algorithms that treat correlated variables as if they were independent will necessarily underestimate their covariance. Underestimated covariance can lead to divergence and make data association extremely difficult. Ulmann and Juiler present an alternative to maintaining the complete joint covariance matrix called Covariance Intersection [45]. Covariance Intersection updates the landmark position variances conservatively, in such a way that allows for all possible correlations between the observation and the landmark. Since the correlations between landmarks no longer need to be maintained, the resulting SLAM algorithm requires linear time and memory. Unfortunately, the landmark estimates tend to be extremely conservative, leading to extremely slow convergence and highly ambiguous data association.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Graphical Optimization Methods

There exists a family of off-line SLAM algorithms which treat the SLAM problems as optimization problem. These techniques exploit a similar independence as discussed in the previous chapter. By retaining all past poses, the set of constraints between different variables in the SLAM posterior can be represented by a set of sparse links. Relaxing those links using conventional optimization techniques then leads to a most likely map and a most likely robot path; assuming known correspondence. Possibly the earliest work on this paradigm is by Lu and Milios [53], which was first implemented by Gutmann [38]. Golfarelli et al. [33] established the relation of SLAM problems and spring-mass models, and Duckett et al. [24] provided a first efficient technique for solving such problems; see also subsequent work in $[48,58]$. The relation between covariances and the information matrix is discussed in [29] and [15]. Folkesson and Christensen introduced a term “Graphical SLAM” into the literature [28], for a related graphical relaxation method. A state-ofthe-art discussion of graphical optimization techniques can be found in [87]. The optimization-based graphical methods are usually offline, and as such are related to the Structure From Motion literature $[43,91]$.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Robust Data Association

In real SLAM applications, the data associations $n^{t}$ are rarely observable. However, if the uncertainty in landmark positions is low relative to the average distance between landmarks, simple heuristics for determining the correct data association can be quite effective. In particular, the most common approach to data association in SLAM is to assign each observation using a maximum likelihood rule. In other words, each observation is assigned to the landmark most likely to have generated it. If the maximum probability is below some fixed threshold, the observation is considered for addition as a new landmark.
In the case of the EKF, the probability of the observation can be written as a function of the difference between the observation $z_{t}$ and the expected observation $\hat{z}{n{t}}$. This difference is known as the “innovation.”
\begin{aligned} \hat{n}{t} &=\underset{n{t}}{\operatorname{argmax}} p\left(z_{t} \mid n_{t}, s^{t}, z^{t-1}, u^{t}, \hat{n}^{t-1}\right) \ &=\underset{n_{t}}{\operatorname{argmax}} \frac{1}{\sqrt{\left|2 \pi Z_{t}\right|}} \exp \left{-\frac{1}{2}\left(z_{t}-\hat{z}{n{t}}\right)^{T} Z_{t}^{-1}\left(z_{t}-\hat{z}{n{t}}\right)\right} \end{aligned}
This data association heuristic is often reformulated in terms of negative $\log$ likelihood, as follows:
$$\hat{n}{t}=\underset{n{t}}{\operatorname{argmin}} \ln \left|Z_{t}\right|+\left(z_{t}-\hat{z}{n{t}}\right)^{T} Z_{t}^{-1}\left(z_{t}-\hat{z}{n{t}}\right)$$
The second term of this equation is known as Mahalanobis distance [80], a distance metric normalized by the covariances of the observation and the landmark estimate. For this reason, data association using this metric is often referred to as “nearest neighbor” data association [3], or nearest neighbor gating.

Maximum likelihood data association generally works well when the correct data association is significantly more probable than the incorrect associations. However, if the uncertainty in the landmark positions is high, more than one data association will receive high probability. If a wrong data association is picked, this decision can have a catastrophic result on the accuracy of the resulting map. This kind of data association ambiguity can be induced easily if the robot’s sensors are very noisy.

One approach to this problem is to only incorporate observations that lead to unambiguous data associations (i.e. if only one data association falls within the nearest neighbor threshold). However, if the SLAM environment is noisy, a large percentage of the observations will go unprocessed. Moreover, failing to incorporate observations will lead to overestimated landmark covariances, which makes future data associations even more ambiguous.

A number of more sophisticated approaches to data association have been developed in order to deal with ambiguity in noisy environments.

## robotics代写|SLAM定位算法代写Simultaneous Localization and Mapping|Robust Data Association

\begin{aligned} \hat{n}{t} &=\underset{n{t}}{\operatorname{argmax}} p\left(z_{t} \mid n_{t}, s^{t} , z^{t-1}, u^{t}, \hat{n}^{t-1}\right) \ &=\underset{n_{t}}{\operatorname{argmax}} \frac{ 1}{\sqrt{\left|2 \pi Z_{t}\right|}} \exp \left{-\frac{1}{2}\left(z_{t}-\hat{z}{n {t}}\right)^{T} Z_{t}^{-1}\left(z_{t}-\hat{z}{n{t}}\right)\right} \end{aligned}\begin{aligned} \hat{n}{t} &=\underset{n{t}}{\operatorname{argmax}} p\left(z_{t} \mid n_{t}, s^{t} , z^{t-1}, u^{t}, \hat{n}^{t-1}\right) \ &=\underset{n_{t}}{\operatorname{argmax}} \frac{ 1}{\sqrt{\left|2 \pi Z_{t}\right|}} \exp \left{-\frac{1}{2}\left(z_{t}-\hat{z}{n {t}}\right)^{T} Z_{t}^{-1}\left(z_{t}-\hat{z}{n{t}}\right)\right} \end{aligned}

n^吨=精氨酸n吨ln⁡|从吨|+(和吨−和^n吨)吨从吨−1(和吨−和^n吨)

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